Analysis of burst activity of the buccal ganglion of Aplysia depilans

1976 ◽  
Vol 64 (2) ◽  
pp. 385-404
Author(s):  
R. M. Rose
Keyword(s):  
Burst Activity ◽  
Network Activity ◽  
Van Der Pol Equation ◽  
Single Cycle ◽  
Buccal Ganglion ◽  
Van Der Pol ◽  
Nonlinear Relaxation ◽  
Nonlinear Properties ◽  
Relaxation Oscillators ◽  
Inhibitory Components

1. Certain nonlinear properties of molluscan neurones suggest that network activity could be described in relation to nonlinear relaxation oscillators. 2. A specific example is fitted by the Van der Pol equation with an exponentially decaying damping coefficient, and changes in the associated energy and sensitivity are considered. 3. More precise details of the interaction pattern determined by analysis of a single cycle of burst activity, and it is shown that there are two states of activity, depending on the balance of presumed inhibitory components. 4. Previous results are discussed in relation to an overall decaying oscillation. 5. Changes in subcomponents of a cycle are described for decaying sequences and the constancy of synaptic interaction is demonstrated. The results are briefly discussed in relation to catastrophe theory and learning.

scholarly journals On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

2014 ◽  
Vol 24 (05) ◽  
pp. 1450061 ◽  
Author(s):  
Albert D. Morozov ◽  
Olga S. Kostromina
Keyword(s):  
Saddle Point ◽  
Limit Cycles ◽  
Small Neighborhood ◽  
Integrable Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Autonomous Equation ◽  
Nonautonomous Equation ◽  
Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

scholarly journals Study of fractional order Van der Pol equation

2016 ◽  
Vol 28 (1) ◽  
pp. 55-60 ◽  
Author(s):  
V. Mishra ◽  
S. Das ◽  
H. Jafari ◽  
S.H. Ong
Keyword(s):  
Fractional Order ◽  
Van Der Pol Equation ◽  
Van Der Pol

scholarly journals A generalization of Van der Pol equation

2013 ◽  
Vol 8 ◽  
pp. 1723-1726
Author(s):  
Ana-Magnolia Marin-Ramirez ◽  
Ruben-Dario Ortiz-Ortiz ◽  
Joel-Arturo Rodriguez-Ceballos
Keyword(s):  
Van Der Pol Equation ◽  
Van Der Pol

Non-standard reduction of noisy Duffing—van der Pol equation

2001 ◽  
Vol 16 (3) ◽  
pp. 223-245 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Richard B. Sowers ◽  
Lalit Vedula
Keyword(s):  
Van Der Pol Equation ◽  
Van Der Pol
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